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Properties

Label 2-231-77.76-c1-0-1
Degree $2$
Conductor $231$
Sign $0.999 + 0.0403i$
Analytic cond. $1.84454$
Root an. cond. $1.35814$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.30i·2-s + i·3-s − 3.33·4-s + 3.77i·5-s + 2.30·6-s + (0.474 + 2.60i)7-s + 3.07i·8-s − 9-s + 8.71·10-s + (3.23 + 0.726i)11-s − 3.33i·12-s − 3.32·13-s + (6.01 − 1.09i)14-s − 3.77·15-s + 0.441·16-s + 5.12·17-s + ⋯
L(s)  = 1  − 1.63i·2-s + 0.577i·3-s − 1.66·4-s + 1.68i·5-s + 0.942·6-s + (0.179 + 0.983i)7-s + 1.08i·8-s − 0.333·9-s + 2.75·10-s + (0.975 + 0.219i)11-s − 0.962i·12-s − 0.923·13-s + (1.60 − 0.293i)14-s − 0.974·15-s + 0.110·16-s + 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(231\)    =    \(3 \cdot 7 \cdot 11\)
Sign: $0.999 + 0.0403i$
Analytic conductor: \(1.84454\)
Root analytic conductor: \(1.35814\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{231} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 231,\ (\ :1/2),\ 0.999 + 0.0403i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11518 - 0.0225170i\)
\(L(\frac12)\) \(\approx\) \(1.11518 - 0.0225170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (-0.474 - 2.60i)T \)
11 \( 1 + (-3.23 - 0.726i)T \)
good2 \( 1 + 2.30iT - 2T^{2} \)
5 \( 1 - 3.77iT - 5T^{2} \)
13 \( 1 + 3.32T + 13T^{2} \)
17 \( 1 - 5.12T + 17T^{2} \)
19 \( 1 + 2.74T + 19T^{2} \)
23 \( 1 - 2.19T + 23T^{2} \)
29 \( 1 + 4.27iT - 29T^{2} \)
31 \( 1 - 1.11iT - 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 - 3.75T + 41T^{2} \)
43 \( 1 + 6.74iT - 43T^{2} \)
47 \( 1 + 5.58iT - 47T^{2} \)
53 \( 1 - 6.66T + 53T^{2} \)
59 \( 1 - 7.36iT - 59T^{2} \)
61 \( 1 - 9.23T + 61T^{2} \)
67 \( 1 - 1.96T + 67T^{2} \)
71 \( 1 + 5.07T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 - 5.88iT - 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 + 10.0iT - 89T^{2} \)
97 \( 1 + 14.4iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.71867023213028657534438868363, −11.41919243297468682683793932381, −10.22464278828909167824037109018, −9.863484647145005110023967421527, −8.779065577438033045578450198073, −7.19323436196514110416496343816, −5.82294449846260798833219434532, −4.24486432994936166363237591112, −3.11523139324577675015833526664, −2.27526022097897077730158167410, 1.01674134541080024853234932960, 4.22728449095516793880842039170, 5.10128038078901837450168410784, 6.16516804130838174400910107890, 7.31313527291810955387033951910, 7.996951721659073059784069548885, 8.878316785490553120617032791667, 9.738391522463962439116919193979, 11.52006718733132445065315585687, 12.63728297752719880770272783752

Graph of the $Z$-function along the critical line